The case where \(H\)  is a Hopf algebra was studied in detail in [1]}.
Any right ideal \(Q\subseteq H^+\)  of the augmentation ideal \(H^+:=\ker \epsilon \) 
that is invariant under the right adjoint \(H\) -coaction \(\mathrm {ad}\colon H^+\rightarrow H^+\otimes H\) , \(h\mapsto h{_{\underline{2}}} \otimes S(h {_{\underline{1}}} ) h{_{\underline{3}}}\) ,
i.e., \(\mathrm {ad}(Q) \subseteq Q\otimes H\) , induces a bicovariant differential
calculus on \(H\) . The corresponding \(H\) -subbimodule \(N_H\subseteq \Gamma ^1(H)\) 
is generated by the image of \(Q\subseteq H^+\)  under the linear map
\(\kappa := (S\otimes \mathrm {id})\circ \Delta \colon H^+ \rightarrow \Gamma ^1(H)\) .
The quotient right \(H\) -comodule
\(\mathfrak {h}^\vee _Q := H^+ / Q\) 
